Suppose x $$\left[ n \right]$$ is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at z = ± 2j. Which one of the following statements is TRUE for the signal x=$$\left[ n \right]$$ ?

The z-transform of the sequence x$$\left[ n \right]$$ is given by x(z)= $${1 \over {{{(1 - 2{z^{ - 1}})}^2}}}$$ , with the region of convergence $$\left| z \right| > 2$$. Then, $$x\left[ 2 \right]$$ is ____________________.

Let $${H_1}(z) = {(1 - p{z^{ - 1}})^{ - 1}},{H_2}(z) = {(1 - q{z^{^{ - 1}}})^{ - 1}}$$ , H(z) =$${H_1}(z)$$ +r $${H_2}$$. The quantities p, q, r are real numbers. Consider , p=$${1 \over 2}$$, q=-$${1 \over 4}$$ $$\left| r \right|$$ <1. If the zero H(z) lies on the unit circle, the r = ____________________________.

The input-output relationship of a causal stable LTI system is given as
𝑦[𝑛] = 𝛼 𝑦[𝑛 − 1] + $$\beta $$ x[n].
If the impulse response h[n] of this system satisfies the condition $$\sum\limits_{n = 0}^\infty h $$[n] = 2, the relationship between α and is $$\alpha $$ and $$\beta $$ is